| L(s) = 1 | + (−2.53 − 6.11i)2-s + (7.02 − 4.69i)3-s + (−19.6 + 19.6i)4-s + (−21.1 − 4.20i)5-s + (−46.5 − 31.0i)6-s + (70.6 − 14.0i)7-s + (71.9 + 29.7i)8-s + (−3.64 + 8.79i)9-s + (27.8 + 139. i)10-s + (116. − 174. i)11-s + (−45.7 + 230. i)12-s + (56.1 + 56.1i)13-s + (−264. − 396. i)14-s + (−168. + 69.7i)15-s − 70.7i·16-s + (−185. + 221. i)17-s + ⋯ |
| L(s) = 1 | + (−0.632 − 1.52i)2-s + (0.781 − 0.521i)3-s + (−1.22 + 1.22i)4-s + (−0.845 − 0.168i)5-s + (−1.29 − 0.863i)6-s + (1.44 − 0.286i)7-s + (1.12 + 0.465i)8-s + (−0.0449 + 0.108i)9-s + (0.278 + 1.39i)10-s + (0.961 − 1.43i)11-s + (−0.318 + 1.59i)12-s + (0.332 + 0.332i)13-s + (−1.35 − 2.02i)14-s + (−0.748 + 0.309i)15-s − 0.276i·16-s + (−0.643 + 0.765i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.659i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.752 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.364380 - 0.968786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364380 - 0.968786i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (185. - 221. i)T \) |
| good | 2 | \( 1 + (2.53 + 6.11i)T + (-11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (-7.02 + 4.69i)T + (30.9 - 74.8i)T^{2} \) |
| 5 | \( 1 + (21.1 + 4.20i)T + (577. + 239. i)T^{2} \) |
| 7 | \( 1 + (-70.6 + 14.0i)T + (2.21e3 - 918. i)T^{2} \) |
| 11 | \( 1 + (-116. + 174. i)T + (-5.60e3 - 1.35e4i)T^{2} \) |
| 13 | \( 1 + (-56.1 - 56.1i)T + 2.85e4iT^{2} \) |
| 19 | \( 1 + (-101. - 244. i)T + (-9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-288. - 192. i)T + (1.07e5 + 2.58e5i)T^{2} \) |
| 29 | \( 1 + (102. - 516. i)T + (-6.53e5 - 2.70e5i)T^{2} \) |
| 31 | \( 1 + (489. + 731. i)T + (-3.53e5 + 8.53e5i)T^{2} \) |
| 37 | \( 1 + (-326. + 218. i)T + (7.17e5 - 1.73e6i)T^{2} \) |
| 41 | \( 1 + (418. - 83.2i)T + (2.61e6 - 1.08e6i)T^{2} \) |
| 43 | \( 1 + (489. - 1.18e3i)T + (-2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (2.19e3 + 2.19e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (45.4 + 109. i)T + (-5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-2.45e3 - 1.01e3i)T + (8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (-1.58 - 7.98i)T + (-1.27e7 + 5.29e6i)T^{2} \) |
| 67 | \( 1 - 338. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (2.58e3 - 1.72e3i)T + (9.72e6 - 2.34e7i)T^{2} \) |
| 73 | \( 1 + (-748. - 148. i)T + (2.62e7 + 1.08e7i)T^{2} \) |
| 79 | \( 1 + (-5.18e3 + 7.76e3i)T + (-1.49e7 - 3.59e7i)T^{2} \) |
| 83 | \( 1 + (7.63e3 - 3.16e3i)T + (3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (6.79e3 - 6.79e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (1.13e3 - 5.68e3i)T + (-8.17e7 - 3.38e7i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.25720355919934278011753844846, −16.80007554663986064284856849781, −14.55639593807566624352184474678, −13.33997963532959354316624208637, −11.63668067146667145889939950681, −11.03751209937021190727661996002, −8.762786583396010922777622155270, −8.057635712374429790837186684579, −3.76252177230675619862891585959, −1.50159548599575858979574658307,
4.64421759902483005840908327671, 7.11551675606306503538555467494, 8.345638013846425177132776780166, 9.388770713066914034405697967981, 11.62075353898678619832097406871, 14.28905607182939030098302497052, 15.04762327841291976160287728470, 15.62696378382394691021484428139, 17.38784674927043762318854410189, 18.12253301455962985344822924170
